Dans ce travail, nous présentons une nouvelle caractérisation de la norme des espaces de Besov-Orlicz associés à la $𝒩$-fonction exponentielle $M_{\beta }$ pour $\beta \>0$. Nous utilisons cette nouvelle norme et un lemme de Marcus et Pisier [15], pour démontrer un critère de tension et de régularité dans les espaces de Besov-Orlicz pour $\beta \ge 1$. Nous étudions ensuite dans les espaces de Besov-Orlicz pour $\beta =1$, des théorèmes limites pour les mesures d’occupations du temps local du processus stable symétrique d’indice $1\<\alpha \le 2$, ce qui...

We investigate in this paper the properties of some dilatations or contractions of a sequence (αn)n≥1 of Lr-optimal quantizers of an ${ℝ}^d$-valued random vector $X\in L^r\left(\mathbb{P}\right)$ defined in the probability space $(\Omega ,𝒜,\mathbb{P})$ with distribution ${\mathbb{P}}_X=P$. To be precise, we investigate the Ls-quantization rate of sequences ${\alpha }_n^{\theta ,\mu }=\mu +\theta ({\alpha }_n-\mu )=\{\mu +\theta (a-\mu ),a\in {\alpha }_n\}$ when $\theta \in {ℝ}_{+}^{☆},\mu \in ℝ,s\in (0,r)$ or s ∈ (r, +∞) and $X\in L^s\left(\mathbb{P}\right)$. We show that for a wide family of distributions, one may always find parameters (θ,µ) such that (αnθ,µ)n≥1 is Ls-rate-optimal. For the Gaussian and the exponential distributions we show...

We consider the one-sided exit problem – also called one-sided barrier problem – for ($\alpha$-fractionally) integrated random walks and Lévy processes. Our main result is that there exists a positive, non-increasing function $\alpha \mapsto \theta \left(\alpha \right)$ such that the probability that any $\alpha$-fractionally integrated centered Lévy processes (or random walk) with some finite exponential moment stays below a fixed level until time $T$ behaves as $T^{-\theta \left(\alpha \right)+\mathrm{o}\left(1\right)}$ for large $T$. We also investigate when the fixed level can be replaced by a different barrier...